Sketch the function [tex]f(x)=-2(x+5)^2+6, x \geq-5[/tex], and its inverse on the same graph.
All graphs must be done by hand. No graphing software. Use different colours for each curve.
d) State the domain and range of the function and its inverse using set notation:
| | [tex]f(x)[/tex] | Inverse |
|------------|----------------|---------|
| Domain | | |
| Range | | |
Answer (2)
The domain and range of the function and its inverse are:
For f ( x ) :
Domain: x ∈ R : x ≥ − 5 Range: y ∈ R : y ≤ 6
For the inverse: Domain: x ∈ R : x ≤ 6 Range: y ∈ R : y ≥ − 5
Examples
Understanding inverse functions is crucial in many real-world applications. For instance, consider converting temperature from Celsius to Fahrenheit using the formula F = 5 9 C + 32 . The inverse function, C = 9 5 ( F − 32 ) , allows you to convert Fahrenheit back to Celsius. This concept is also used in cryptography, where encryption and decryption are inverse processes, ensuring secure communication. Moreover, in economics, demand and supply curves are often inverses of each other, helping to analyze market equilibrium.
The function f ( x ) = − 2 ( x + 5 ) 2 + 6 has a domain of { x ∈ R : x ≥ − 5 } and a range of { y ∈ R : y ≤ 6 } . Its inverse, f − 1 ( x ) = − 5 + 2 6 − x , has a domain of { x ∈ R : x ≤ 6 } and a range of { y ∈ R : y ≥ − 5 } . Both functions can be sketched on the same graph showing their reflection across the line y = x .
;
